Normalized solutions for nonlinear Schrödinger systems with special mass-mixed terms: The linear couple case

Abstract

In this paper, we prove the existence of positive solutions $(λ_1,λ_2, u,v)\in \R^2\times H^1(\R^N, \R^2)$ to the following coupled Schrödinger system $$\begin{cases} -Δu + λ_1 u= μ_1|u|^{p-2}u+βv \quad &\hbox{in}\;\RN, \\ -Δv + λ_2 v= μ_2|v|^{q-2}v+βu \quad &\hbox{in}\;\RN, \end{cases}$$ satisfying the normalization constraints $\displaystyle\int_{\RN}u^2 =a, ~ \int_{\RN}v^2 =b$. The parameters $μ_1,μ_2,β>0$ are prescribed and the masses $a,b>0$. Here $2+\frac{4}{N} 0$ and $β>0$, provided $2\leqslant N\leqslant 4$. For the Sobolev critical case with $N=3,4$, it can be viewed as a counterpart of the Brezis-Nirenberg critical semilinear elliptic problem for the system case in the context of normalized solutions. Under some suitable assumptions, we obtain the existence or non-existence of positive normalized ground state solution.